Journal of Sound and Vibration (1997) 205(5), 617–630

VIBRATION ANALYSIS OF CIRCULAR MINDLINPLATES USING THE DIFFERENTIAL

QUADRATURE METHOD

K. M. L, J.-B. H Z. M. X

Division of Engineering Mechanics, School of Mechanical and Production Engineering,Nanyang Technological University, Nanyang Avenue, Singapore 639798

(Received 10 April 1996, and in final form 11 March 1997)

Axisymmetric free vibrations of moderately thick circular plates described by the linearshear-deformation Mindlin theory are analyzed by the differential quadrature (DQ)method. The first fifteen natural frequencies of vibration are calculated for uniform circularplates with free, simply-supported and clamped edges. Through these computations, thecapability and simplicity of the differential quadrature method for moderately thick plateeigenvalue analysis is demonstrated, and convergence and accuracy are thoughtfullyexamined. The case of a rigid point support at the plate centre is also considered in thepresent paper, for which special attention is paid to the capability and convergence of thecurrent method.

7 1997 Academic Press Limited

1. INTRODUCTION

As kinds of basic structural components, thin and moderately thick circular plates areextensively used in mechanical, civil, nuclear and aerospace structures. Considerablestudies have been reported in the open literature on the axisymmetric vibration analysisof circular plates, e.g., [1–7]. Excellent reviews have been made by Leissa [8–14]. Recently,the differential quadrature (DQ) method [15–19] has been applied to free vibration analysisof circular and annular thin plates with uniform or non-uniform thickness [20–24]; theseapplications concentrate mainly on the fundamental frequency.

The differential quadrature method is a rather efficient numerical method for the rapidsolution of linear and non-linear partial differential equations. It was originated byBellman and Casti [15] and Bellman et al. [16], and generalized and simplified further byQuan and Chang [17, 18] and Shu and Richards [19] through introducing simple algebraicexpressions to calculate directly the weighting coefficients associated with derivatives.Thanks to the efforts of Bert et al. [25, 26], Striz et al. [27], Sherbourne and Pandey [28],and Kukreti et al. [29], the method is becoming increasingly popular in the solution ofbending, buckling and free vibration problems of structures. In all the aforementionedstudies [15–29], the DQ method appears to be a potential alternative to the conventionalnumerical approaches, and has been claimed to have the capability of yielding highlyaccurate solutions to initial and boundary value problems with minimal computationaleffort.

In view of the fact that no publications are concerned with the free vibration analysisof moderately thick plates using the DQ method, in this paper, the method is employedto analyze the axisymmetric free vibration of moderately thick circular Mindlin plates withfree, simply-supported and clamped edges. The first fifteen natural frequencies of the plates

0022–460X/97/350617+14 $25.00/0/sv971035 7 1997 Academic Press Limited

z

a

hr

. . .618

are calculated. The capability and simplicity of the DQ method in moderately thick plateeigenvalue analysis are demonstrated through these studies. The accuracy and convergenceof the method for the vibration analysis of moderately thick plates are investigated throughdirectly comparing DQ results with corresponding exact solutions in the open literature.A particular advantage involved in the present solution procedures, i.e., using thesimplified version of the DQ method along with the Mindlin plate theory, is noted. Thecase of a rigid point support at the plate centre is also included in the present investigation.The capability and convergence characteristics of the method for this particular problemare explored.

2. MATHEMATICAL FORMULATIONS

2.1.

Consider a circular plate of radius a and thickness h (Figure 1). The equations governingthe axisymmetric free vibration of a uniform circular plate of isotopic material can bederived using Mindlin’s theory [30] as [31, 32]:

D012c

1r2 +1r

1c

1r−

c

r21− kGh01w1r

+c1−rh3

1212c

1t2 =0, (1a)

kGh012w1r2 +

1r

1w1r

+1c

1r+

c

r1− rh12w1t2 =0, (1b)

where w is the transverse deflection; c is the rotation of the normal about the r-axis;D=Eh3/[12(1− n2)], E, G and n are the plate flexural rigidity, Young’s modulus, shearmodulus and Poisson’s ratio, respectively; r and k are the density of the plate materialand the shear correction factor respectively.

According to the relationship between force resultants and deformation variables, thefollowing formulae are obtained:

Mr =D01c

1r+ n

c

r1; Mu =D0n 1c

1r+

c

r1; Qr = kGh01w1r

+c1, (2)

in which Mr , Mu and Qr are the moment resultants and the shear resultant.Using the following non-dimensional parameters and relation:

R= r/a; d= h/a; W=w/a; c=c; T= tzE/ra2(1− n2); (3)

Figure 1. Configuration of a circular thick plate.

619

the governing equations given by equation (1) can be normalized as:

d20R2 12C

1R2 +R1C

1R−C1−6k(1− n)R20C+

1W1R1 − d2R2 12C

1T2 =0, (4a)

R12W1R2 +

1W1R

+R1C

1R+C−

2R(1− n)k

12W1T 2 =0, (4b)

and the stress–displacement relationships are given by:

Mr =Da $1C

1R+

n

RC%; Mu =

Da $n 1C

1R+

C

R%; Qr = kGh$C+1W1R%. (5)

For free vibration, the solution can be assumed as:

W(R, T)=Wj (R)eiVjT and C(R, T)=Cj (R)eiVjT. (6)

Substitution of these solutions into the homogeneous differential equations leads to

d20R2 d2C

dR2 +RdC

dR−C1−6k(1− n)R20C+

dWdR1+ d2R2V2C=0, (7a)

Rd2WdR2 +

dWdR

+RdC

dR+C+

2RV2

(1− n)kW=0, (7b)

W, C and V, should have been taken as Wj (R), Cj (R) and Vj of equation (6), respectivelyfor the jth mode of vibration. Here and in the following, the suffix j is dropped for thesake of convenience.

According to the DQ procedure (refer to the Appendix for details) and by setting R1 =0and RN =1, equations (7) take the following discrete forms:

d2 sN

k=1

(C(2)ik R2

i +C(1)ik Ri )Ck −[d2 +6k(1− n)R2

i ]Ci

−6k(1− n)R2i s

N

k=1

C(1)ik Wk + d2R2

i V2Ci =0, (8a)

sN

k=1

(C(2)ik Ri +C(1)

ik )Wk +Ci +Ri sN

k=1

C(1)ik Ck +

2Ri

(1− n)kV2Wi =0, (8b)

where i=1, 2, . . . , N. C(n)rs , determined by equations (A2)–(A5), are the weighting

coefficients for the nth order derivatives of W and C with respect to R.

2.2.

For the edge of the circular plate, the boundary conditions can be divided into thefollowing three kinds:

(1) Clamped edge (C): w=0; c=0, (9)

(2) Simply–supported edge (S): w=0; Mr =0, (10)

(3) Free edge (F): Qr =0; Mr =0. (11)

. . .620

These conditions can be further expressed as:

(C) W=0; C=0, (12)

(S) W=0; 1C/1R+(n/R)C=0, (13)

(F) C+ 1W/1R=0; 1C/1R+(n/R)C=0, (14)

Thus, the discretized forms on the edge of the plate are:

(C) WN =0; CN =0, (15)

(S) WN =0; sN

k=1

C(1)NkCk + nCN =0, (16)

(F) CN + sN

k=1

C(1)NkWk =0; s

N

k=1

C(1)NkCk + nCN =0. (17)

For the central point (r=0), the restraint conditions and their discretized forms are:

(4) Regularity conditions (R) Qr =0; c=0. (18)

(5) Rigid centre support (C) w=0; c=0. (19)

and

(R) C1 + sN

k=1

C(1)1k Wk =0; C1 =0, (20)

(C) W1 =0; C1 =0. (21)

2.3.

Since both the discretized governing equations and the discretized constraint conditionsare written out on a point-wise basis, for the boundary and centre points, the discretizedgoverning equations and the discretized constraint conditions should be satisfiedsimultaneously. In order to get solutions of the problem, however, one has to use thediscretized constraint conditions instead of the discretized governing equations on both theboundary and the centre points. Thus, the solutions of the problems are acquired bysolving the set of secular equations which consists of 2× (N−2) governing equations atall the non-boundary/centre points and 2×2 constraint conditions at both the edge andcentre points.

When using the differential quadrature method together with thin plate theory, adifficulty in dealing with boundary constraints arises. The reason is that there is only onegoverning equation but two boundary conditions which should be satisfied at eachboundary point. To overcome this difficulty, the d-method has been introduced [33] inwhich the two respective boundary conditions are applied both at the boundary and ata very small distance d from the boundary. However, due to the DQM being a polynomialapproach [15–17], applying boundary conditions on non-boundary points will cause thepolynomial to oscillate, and these oscillations will be strengthened with the increasing orderof the approximating polynomial (i.e., by increasing the number of grid points employed).This problem is naturally evaded by using the Mindlin plate theory, in which there aretwo governing equations and two constraint conditions at both central and edge pointsfor axisymmetric problems.

(a) (d)

(b) (e)

(c) (f)

621

3. RESULTS AND DISCUSSION

Based on the formulations presented in the previous section, a program has beendeveloped. For simplicity, only uniform thickness plates are considered in the presentstudy; the Poisson’s ratio is taken as n=0·3, and the shear correction factor k is takenas p2/12 [30]. The grid points employed in the computations are designated by:

Ri =12 $1−cos 0(i−1)p

N−1 1%, i=1, 2, . . . , N. (22)

A non-dimensional frequency parameter l2 is adopted for the results presentation, and isdefined as:

l2 =va2zrh/D and v=VzE/ra2(1− n2). (23)

3.1.

In this sub-section, the free vibration analyses of circular Mindlin plates subject tocompletely free, simply-supported and clamped boundary conditions without a rigid pointsupport at the plate centre (Figure 2a, 2b and 2c) are carried out.

In accordance with previous experience [34], convergence and accuracy studies must becarried out to reveal the convergence characteristics of the differential quadrature methodfor a particular problem as well as to ensure accuracy of the results. Thus, in Figure 3,the normalized frequency parameters l2/l2

ext of the first four mode sequences are presentedwith an increasing number of grid points for the completely free circular plates ofh/a=0·001 (Figure 3a) and h/a=0·250 (Figure 3b). Here, the values l2

ext are the exactsolutions taken from [3]. For the simply-supported and clampled circular plates, similarconvergence and accuracy studies are conducted, which are described in Figures 4 and 5.In Tables 1–3, the first fifteen non-dimensional frequency parameters l2, are tabulated forthe various relative thickness plates subject to the free, simply-supported and clampedboundary conditions, respectively. There, the minimum numbers of grid points requiredfor achieving convergent results with five significant digits are also exhibited for the firstfifteen non-dimensional frequency parameters of various relative thickness plates withdifferent boundary conditions. From these figures and tables, the following remarks on theconvergence characteristics and the accuracy of the method for the present problem can

Figure 2. Various circular plates analysed: (a) completely free plate; (b) simply-supported plate; (c) clampedplate; (d) simply-supported plate with a rigid centre support; (e) clamped plate with a rigid centre support; and(f) free plate with a rigid centre support.

15Number of grid points, N

5

1.0

97

Mode 1

Mode 3

Mode 4

11

(a)1.3

1.2

2Mode 2

0.8

1.1N

orm

aliz

ed f

requ

ency

par

amet

er, λ

2 /λex

t

135

1.0

97

Mode 1

Mode 3

Mode 4

11

(b)1.8

1.4

Mode 21.2

1.6

13

0.9

. . .622

Figure 3. Convergence and accuracy of the normalized frequency parameter, l2/l2ext , for the first four mode

sequences with grid refinement for the free circular plates: (a) h/a=0·001 and (b) h/a=0·250. l2ext—the exact

solution [3].

be made: (1) When increasing the number of grid points, the DQ results converge to thecorresponding exact solutions, which is true for all three kinds of boundary conditionsconsidered and for various natural frequencies (at least for the first fifteen naturalfrequencies). (2) Whatever the relative thickness of a plate, the convergence of the DQresults with grid refinement demonstrates a fluctuation characteristic for the free andsimply-supported plates, while for the clamped plate, the frequency parameters obtained

T 1

Convergent results† of frequency parameters, l2 =va2(rh/D)1/2, for free circular plates‡

h/aMode ZXXXXXXXXXXXXXXXXXCXXXXXXXXXXXXXXXXXV

sequences 0·001 0·050 0·100 0·150 0·200 0·250

1 9·0031 (10) 8·9686 (11) 8·8679 (10) 8·7095 (10) 8·5051 (10) 8·2674 (10)2 38·443 (13) 37·787 (13) 36·401 (12) 33·674 (13) 31·111 (13) 28·605 (13)3 87·749 (16) 84·443 (14) 76·676 (16) 67·827 (14) 59·645 (14) 52·584 (14)4 156·81 (16) 146·76 (15) 126·27 (15) 106·40 (15) 90·645 (15) 76·936 (17)5 245·62 (18) 222·38 (18) 181·46 (16) 146·83 (16) 120·57 (16) 99·545 (17)6 354·17 (21) 308·98 (19) 239·98 (21) 187·79 (18) 149·63 (18) 114·53 (16)7 482·45 (22) 404·44 (22) 300·38 (22) 228·39 (20) 171·18 (20) 126·34 (17)8 630·46 (25) 506·96 (23) 361·73 (23) 267·32 (22) 183·36 (22) 138·59 (19)9 798·19 (28) 615·01 (26) 423·41 (24) 297·08 (22) 199·04 (22) 154·77 (20)

10 985·65 (29) 727·37 (27) 484·93 (27) 310·03 (24) 217·13 (24) 166·06 (20)11 1192·8 (30) 843·04 (30) 545·74 (28) 330·92 (24) 231·82 (25) 182·35 (23)12 1419·7 (31) 961·25 (31) 604·75 (29) 351·70 (25) 251·78 (26) 197·61 (23)13 1666·3 (32) 1081·4 (31) 653·91 (32) 372·16 (25) 268·68 (25) 208·73 (24)14 1932·6 (33) 1202·9 (33) 667·41 (30) 397·54 (30) 285·12 (26) 228·95 (25)15 2218·6 (36) 1325·5 (34) 695·93 (30) 416·62 (30) 308·16 (28) 238·48 (26)

† Convergent results with five significant digits.‡ A number in parentheses refers to the minimum number of grid points needed to obtain the convergent resultwith five significant digits.

Nor

mal

ized

fre

quen

cy p

aram

eter

, λ2 /λ

ext

15Number of grid points, N

5

1.0

97

Mode 1

Mode 3

Mode 4

11

(a)1.4

1.22

Mode 2

0.9

1.1

1.3

135

1.0

97

Mode 1

Mode 3

Mode 4

11

(b)1.5

1.2Mode 2

0.9

1.1

1.3

13

1.4

623

Figure 4. Convergence and accuracy of the normalized frequency parameter, l2/l2ext , for the first four mode

sequences with grid refinement for the simply-supported circular plates: (a) h/a=0·001 and (b) h/a=0·250.l2

ext—the exact solution [3].

using the DQ method show essentially monotonic convergence. (3) For various boundaryconditions and relative thicknesses, more grid points are required to acquire a convergentresult for a higher frequency than for a lower one. (4) Using the same number of gridpoints, the thicker a plate (h/a=0·001–0·250), the more accurate the results by the DQmethod will be. When a higher frequency is required, this thickness effect becomes more

T 2

Convergent results† of frequency parameters, l2 =va2(rh/D)1/2, for simply-supportedcircular plates‡

h/aMode ZXXXXXXXXXXXXXXXXXCXXXXXXXXXXXXXXXXXV

sequences 0·001 0·050 0·100 0·150 0·200 0·250

1 4·9351 (9) 4·9247 (8) 4·8938 (8) 4·8440 (8) 4·7773 (8) 4·6963 (8)2 29·720 (11) 29·323 (11) 28·240 (11) 26·715 (11) 24·994 (9) 23·254 (9)3 74·155 (14) 71·756 (14) 65·942 (14) 59·062 (12) 52·514 (12) 46·775 (12)4 138·31 (16) 130·35 (14) 113·57 (15) 96·775 (15) 82·766 (16) 71·603 (15)5 222·21 (18) 202·81 (18) 167·53 (16) 136·98 (15) 113·87 (16) 96·609 (16)6 325·83 (19) 286·79 (19) 225·34 (17) 178·23 (17) 145·13 (16) 108·27 (14)7 499·18 (22) 380·13 (20) 285·44 (20) 219·86 (20) 166·29 (18) 121·50 (16)8 592·27 (23) 480·94 (23) 346·83 (21) 261·51 (20) 176·28 (18) 131·65 (14)9 755·08 (26) 587·65 (24) 408·91 (23) 291·55 (20) 191·38 (18) 146·17 (18)

10 937·61 (27) 698·97 (27) 471·31 (25) 303·05 (22) 207·23 (19) 163·30 (15)11 1139·9 (28) 813·85 (30) 533·80 (28) 318·34 (21) 227·28 (20) 170·65 (18)12 1361·8 (29) 931·50 (29) 596·23 (27) 344·39 (24) 237·98 (22) 194·94 (21)13 1603·5 (30) 1051·2 (31) 649·29 (30) 359·27 (22) 268·49 (24) 198·98 (21)14 1864·9 (32) 1172·6 (31) 658·55 (30) 385·53 (25) 269·03 (24) 219·11 (23)15 2145·9 (35) 1295·1 (32) 677·58 (27) 408·98 (23) 298·91 (26) 236·92 (20)

† Convergent results with five significant digits.‡ A number in parentheses refers to the minimum number of grid points needed to obtain the convergent resultwith five significant digits.

13

1.6

Number of grid points, N

Nor

mal

ized

fre

quen

cy p

aram

eter

, λ2 /λ

ext

5

1.0

97

Mode 1

Mode 3

Mode 4

11

(a)

1.4

1.2

2

13

1.6

5

1.0

97

Mode 1

Mode 3

Mode 4

11

(b)

1.4

1.2

Mode 2 Mode 2

. . .624

Figure 5. Convergence and accuracy of the normalized frequency parameter, l2/l2ext , for the first four mode

sequences with grid refinement for the clamped circular plates (a) h/a=0·001 and (b) h/a=0·250. l2ext—the exact

solution [3].

dominant. (5) For all three kinds of boundary conditions considered herein, the DQsolutions of the first fifteen natural frequencies by 36 grid points are convergent with atleast five significant digits; and the DQ solutions of the first four natural frequencies by11 grid points can be regarded as ones with enough accuracy (3–4 significant digits). Inorder to further demonstrate the accuracy of the DQ method, Table 4 introduces acomparison between the DQM results and the exact solutions obtained by Irie et al. [3]

T 3

Convergent results† of frequency parameters, l2 =va2(rh/D)1/2, for clamped circular plates‡

h/aMode ZXXXXXXXXXXXXXXXXXCXXXXXXXXXXXXXXXXXV

sequences 0·001 0·050 0·100 0·150 0·200 0·250

1 10·216 (8) 10·145 (8) 9·9408 (8) 9·6286 (8) 9·2400 (8) 8·8068 (8)2 39·771 (11) 38·855 (11) 36·479 (9) 33·393 (9) 30·211 (9) 27·253 (9)3 89·102 (12) 84·995 (12) 75·664 (12) 65·551 (11) 56·682 (10) 49·420 (11)4 158·18 (13) 146·40 (13) 123·32 (13) 102·09 (12) 85·571 (13) 73·054 (13)5 246·99 (15) 220·73 (15) 176·41 (15) 140·93 (13) 115·55 (16) 97·198 (14)6 355·54 (19) 305·71 (16) 232·97 (16) 180·99 (15) 145·94 (16) 117·90 (14)7 483·82 (20) 399·32 (19) 291·71 (17) 221·62 (19) 174·97 (17) 122·43 (15)8 631·83 (22) 499·82 (20) 351·82 (20) 262·45 (21) 178·76 (17) 144·42 (17)9 799·57 (23) 605·78 (23) 412·77 (22) 301·11 (21) 205·32 (18) 148·75 (17)

10 987·03 (25) 716·07 (24) 474·18 (23) 305·15 (21) 210·53 (18) 170·37 (20)11 1194·2 (25) 829·74 (25) 535·81 (25) 336·52 (22) 237·46 (22) 181·05 (18)12 1421·1 (27) 946·07 (28) 597·43 (26) 345·58 (24) 248·18 (21) 195·12 (21)13 1667·7 (29) 1064·5 (27) 657·60 (27) 380·88 (25) 268·60 (23) 216·40 (23)14 1934·0 (31) 1184·5 (29) 662·37 (27) 388·16 (25) 290·67 (24) 220·58 (22)15 2220·0 (32) 1305·7 (31) 698·63 (28) 425·43 (26) 299·71 (24) 243·02 (24)

† Convergent result with five significant digits.‡ A number in parentheses refers to the minimum number of grid points needed to obtain the convergent resultwith five significant digits.

100

300

0

Number of grid points, N

Fre

quen

cy p

aram

eter

, λ2

= ω

a2 (ρh

/D)1/

2

25

200

100

50

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

75

(a)

100

125

025

100

25

50

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

75

(b)

75

50

625

T 4

Comparison study of frequency parameters, l2 =va2(rh/D)1/2, for circular plates withdifferent boundary conditions

h/a=0·001 h/a=0·250Boundary Mode ZXXXXXCXXXXXV ZXXXXXCXXXXXVconditions sequences DQM Exact [3] DQM Exact [3]

S 1 4·935 4·935 4·696 4·6962 29·720 29·720 23·254 23·2543 74·155 74·156 46·775 46·7754 138·314 138·318 71·603 71·603

C 1 10·216 10·216 8·807 8·8072 39·771 39·771 27·253 27·2533 89·102 89·104 49·420 49·4204 158·180 158·184 73·054 73·054

F 1 9·003 9·003 8·267 8·2672 38·443 38·443 28·605 28·6053 87·749 87·750 52·584 52·5844 156·808 156·818 76·936 76·936

for the first four natural frequencies. It is found that, except for some rounding errors,the DQ results are identical to the corresponding exact solutions.

3.2.

To further explore the capability, convergence and accuracy of the differentialquadrature method for plate vibration problems, free vibrations of circular Mindlin plateswith a rigid point support at the plate centre (Figure 2d, 2e and 2f) are investigated. Itis noted that there is a stress singularity at the centre of such plates due to the rigid centresupport.

In Figures 6 and 7, the variations of the first five frequency parameters forsimply-supported and clamped plates with a rigid centre support via the number of grid

Figure 6. Convergence of the first five frequency parameters with grid refinement for simply-supported plateswith a rigid centre support: (a) h/a=0·001 and (b) h/a=0·250.

50

300

0

Number of grids points, N

Fre

quen

cy p

aram

eter

, λ2

= ω

a2 (ρh

/D)1/

2

14

200

100

3223

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

41

(a)

50

125

014

25

100

75

50

3223

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

41

(b)

. . .626

Figure 7. Convergence of the first five frequency parameters with grid refinement for clamped plates with arigid centre support: (a) h/a=0·001 and (b) h/a=0·250.

points are demonstrated respectively. These figures expose the following convergencecharacteristics of the differential quadrature method for such special problems: (1) Fora simply-supported or clamped plate with a rigid centre support, the DQ resultsconverge to a steady value when increasing the number of grid points. (2) For all thefrequencies considered in the figures, the DQ results for a clamped plate with gridrefinement converge much faster than those for a simply-supported plate with the samerelative thickness. (3) The thicker a plate, the faster the DQ results advance tocorresponding convergent values.

Table 5 presents some results of the first and fifth frequency parameters obtained usingdifferent numbers of grid points for simply-supported and clamped plates of h/a=0·001and h/a=0·250. It is found from the table that, for clamped plates, the results by 50grid points can be regarded as ones with enough accuracy (3–4 significant digits),whereas, for simply-supported plates, a grid with 200 points is required to obtainabout the same accuracy. In order to obtain results with acceptable accuracy (about 2significant digits), 14 and 24 grid points are needed for the clampled plates andsimply-supported plates respectively. It is noticed that, for the simply-supportedplates, even using as many as 200 grid points, one still cannot obtain fully convergentresults.

In Table 5, the thin plate exact solutions for the fundamental frequency [8] are alsopresented for comparison. This comparison shows that the DQ results converge to thecorresponding exact solutions when the number of grid points is increased.

For a free plate with a grid centre support, convergence is completely different fromthose for simply-supported or clamped plates. The variations of the first five frequencyparameters via the grid number are illustrated in Figure 8. It is found from the figure, that,even using as many as 200 grid points, the DQ solutions for the first two mode sequencesdo not display any sign of convergence. In fact, after carefully examining the results, onefinds, that for the first frequency, the DQ results obtained using even numbers of gridpoints converge, with grid refinement, to a given value. When odd numbers of grid pointsare used, the results converge to another value. The authors have also tried to solve for

200

100

5Number of grids points, N

Fre

quen

cy p

aram

eter

, λ2

= ω

a2 (ρh

/D)1/

2

50

75

50

25

0

150100

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

627

T 5

Frequency parameters, l2 =va2(rh/D)1/2, of simply-supported and clamped circular plateswith a rigid centre support calculated by different numbers of grid points†

h/a=0·001 h/a=0·250Boundary ZXXXXXXCXXXXXXV ZXXXXXXXCXXXXXXXVconditions Mode 1 Mode 5 Mode 1 Mode 5

S 13·709 (15) 264·95 (15) 8·4773 (15) 99·104 (15)15·904 (16) 280·49 (16) 7·8702 (16) 98·739 (16)14·133 (23) 266·30 (23) 8·0972 (23) 98·698 (23)15·487 (24) 276·73 (24) 7·7267 (24) 98·500 (24)14·514 (49) 269·22 (49) 7·6207 (49) 98·248 (49)14·113 (50) 273·82 (50) 7·4679 (50) 98·117 (50)14·672 (99) 270·44 (99) 7·3093 (99) 97·982 (99)14·953 (100) 272·60 (100) 7·2416 (100) 97·954 (100)14·740 (199) 271·04 (199) 7·0718 (199) 97·796 (199)14·872 (200) 271·96 (200) 7·0411 (200) 97·784 (200)

Ref. [8] 14·8 — — —C 22·809 (13) 288·56 (13) 12·248 (13) 99·560 (13)

22·714 (14) 291·26 (14) 12·193 (14) 99·538 (14)22·754 (19) 299·31 (19) 11·968 (19) 99·303 (19)22·730 (20) 298·63 (20) 11·934 (20) 99·250 (20)22·740 (29) 298·89 (29) 11·710 (29) 99·027 (29)22·734 (30) 298·74 (30) 11·691 (30) 99·003 (30)22·737 (49) 298·79 (49) 11·446 (49) 98·776 (49)22·736 (50) 298·77 (50) 11·437 (50) 98·766 (50)

Ref. [8] 22·7 — — —

† A number in parentheses refers to the number of grid points with which the DQM result is obtained.

the natural frequencies of a free plate with relative thickness of 0·100 or less using thepresent method. It is found that there are some negative values among the eigenvaluesobtained by solving the corresponding determinant equations. Therefore, it may be

Figure 8. Variation of the first five frequency parameters via the number of grid points for a free plate ofh/a=0·250 with a rigid centre support.

. . .628

concluded that, for a free plate with a rigid centre support, the differential quadraturemethod fails to produce convergent and correct solutions.

4. CONCLUSIONS

In this paper, the differential quadrature method has been applied successfully to solvethe axisymmetric free vibration problem of moderately thick circular Mindlin plates. Free,simply-supported and clamped plates without or with a rigid centre support have beenconsidered in the present study. The applicability, convergence properties and accuracy ofthe present method for moderately thick plate eigenvalue problems have been carefullyexamined.

For plates without a rigid centre support, the first fifteen frequency parameters havebeen calculated for various relative thicknesses and for different boundary conditions. Forsuch plates, the DQ method yields convergent and accurate solutions even for a smallnumber of grid points. The results show that different boundary conditions, relative platethicknesses and mode sequences have significant influences on the convergence propertiesof the method.

For simply-supported and clamped plates with a rigid centre support, the DQ resultsapproach the corresponding correct solutions slowly with increasing grid refinement. Usinga grid of about 25 points, the method can provide solutions with acceptable accuracy(about 2 significant digits). However, for free plates with a rigid centre support, the methodseems to fail to produce convergent and correct solutions.

The two advantages involved in the present solution procedure are: (1) By using theMindlin plate theory, the problem that two grid points are required at each boundary pointis naturally avoided. (2) By using algebraic expressions to calculate the weightingcoefficients, the number of grid points can be greatly increased when necessary.

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APPENDIX: THE DIFFERENTIAL QUADRATURE METHOD

Supposing that there are N grid points along the r-axis with r1, r2, . . . , rN as theco-ordinates, the nth order derivative of f(r) can be expressed discretely at the point ri as:

. . .630

f (n)r (ri )= s

N

k=1

C(n)ik f(rk ); n=1, 2, . . . , N−1, (A1)

where C(n)ij are the weighting coefficients associated with the nth order derivative of f(r) at

the discrete point ri .According to references [17, 19], the weighting coefficients in equation (A1) can be

determined as follows:

C(1)ij =M(1)(ri )/[(ri − rj )M(1)(rj )]; i, j=1, 2, . . . , N, but j$ i, (A2)

where

M(1)(ri )= tN

j=1, j$ i

(ri − rj ), i=1, 2, . . . , N (A3)

and

C(n)ij = n0C(n−1)

ii C(1)ij −

C(n−1)ij

ri − rj1; i, j=1, 2, . . . , N, but j$ i; and n=2, 3, . . . , N−1

(A4)

C(n)ii =− s

N

j=1, j$ i

C(n)ij ; i=1, 2, . . . , N, and n=1, 2, . . . , N−1. (A5)